9.4 PROPERTIES OF THE COMPLEX NUMBER FIELD 321EXAMPLE 2 Use the quadratic formula to find the solution(s) to the
equation 5x2 + 2x + 1 = 0.
Solution The quadratic formula yields roots (&)[-2 f J(z2) - (4)(5)(1)],
which simplifies to ( - *) & (&)Jx, which equals ( -# )f (3)i.
Recall that equation i2 = - 1 was our basis, in Article 9.2, for cohcluding
that the complex number field cannot be ordered. In the next definition we
resume our introduction of notation associated with C.
DEFINITION 2
Let z = x + yi be a complex number. We call the real number x the rdal part
of z, denoted Re(z), while the real number y is called the imaginary part of z,
denoted Im (z).
Note that any complex number z can be expressed z = Re(z) + Im (z)i,
Notice also that both the real and imaginary parts of a complex number
z are real numbers. We can reexpress (a) and (b) of Definition 1 easily in
terms of real and imaginary parts. For (a), two complex numbers are equal
if and only if their respective real and imaginary parts are equal. For (b),
the sum of two complex numbers is the complex number whose real (resp.,
imaginary) part is the sum of the real (resp., imaginary) parts of the given
numbers. A complex number z such that Re (z) = 0 is said to be purely
imaginary (or just imaginary). If Im (z) = 0, we say that the complex number
z is real. It is left for you to verify (Exercise 6) that if the complex numbers
z, and z2 are real with real parts x, and x, respectively, then the sum and
product of z, and 2, in C equal the sum and product, respectively, of x,
and x2 in R. Thus complex numbers that are real, in the sense just defined,
behave algebraically just like real numbers, so that we may regard R and the
subset (z E C (Im (z) = 0) of C as, for all intents and purposes, identical. It
is in this precise sense that R may be thought of as a subset of C.
We now state the property of the complex number system that provides
its closest link to the number systems R and Q.
THEOREM 1
The complex number system (C, +, a) is a field.
Outline of proof Verification of field axioms 1 through 3, 6 through 8, and
11 are left to you in Exercise 7(a). For Axioms 4 and 9, we note that the
complex numbers 0 = 0 + Oi and 1 = 1 + Oi serve as additive and multi-
plicative identities, respectively. For example, if z = x + yi E C, then
z 1 = (x + yi)(l + Oi) = ((x)(l) - (y)(O)) + ((y)(l) + (x)(O))(i) = x + yi =
z, as required. You may complete the formal verification of Axiom 4,
as well as show that -2 = -x - yi is the additive inverse of z = x + yi,
that is, Axiom 5. For Axiom 10, we note that if z = x + yi # 0, then the
complex number (x - yi)/(x2 + y2) serves as the multiplicative inverse of
z, and so may be denoted z- '. The details of this as well are left to you.
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